Lebesgue's Theorem Research Articles

H 1-integrals with respect to some bases

In 1998, Garces, Lee, and Zhao defined the so-called H 1-integral [I.J.L. Garces, P.Y. Lee, and D. Zhao, Moore–Smith limits and the Henstock integral, Real Anal. Exch., 24(1):447–456, 1998/99]. Later on, a full characterization of H 1-integrable functions was given by Maliszewski and the author [A. Maliszewski and P. Sworowski, A characterization of H 1-integrable functions, Real Anal. Exch., 28(1):93–104, 2002/03]. The characterization is in terms of generalized continuity and so is similar to the classical Lebesgue theorem on Riemann integrands. The present paper contributes to the theory with some results of this type for H 1-integral defined with respect to differential bases: namely, the McShane basis, path bases, and so-called free point bases. In the latter two cases, the characterization is partial (necessity condition being provided only).

Set-Valued Lebesgue and Riesz Type Theorems

Abstract In this paper, we continue a previous study concerning different types of pseudo-convergences of sequences of measurable functions with respect to set-valued nonadditive monotonic set functions and we establish some pseudo-versions of Lebesgue and Riesz theorems in the set-valued case. We also characterize some important structural properties of fuzzy multimeasures.

A Lebesgue's type theorem on toroidal graphs and its application to linear coloring

A coloring of a graph <italic>G</italic> is said to be linear if any two color classes induce an acyclic subgraph of maximum degree at most 2. In this paper, we first prove a Lebesgue's type theorem concerning the structure of toroidal graphs. As a consequence of this result, we show that every toroidal graph <italic>G</italic> of girth at least 5 can be 「(△(<italic>G</italic>))/2」+ 4)-linear list colorable, unless △(<italic>G</italic>) = 4 and G has a subgraph containing only faces of degree five of which each is incident with three vertices of degree 3 and two vertices of degree 4. This generalizes and improves some known results.

Some remarks about closed convex curves

We introduce a function w k (θ) for closed convex plane curves, and then prove a geometric inequality involving w k (θ) and the area A enclosed by the curve. As a by-product, we give a new proof of the classical isoperimetric inequality. Finally, we give some properties of convex curves with w k (θ) being constant and pose an open problem motivated by the elegant Blaschke―Lebesgue theorem.

Exceptional points for Lebesgue's density theorem on the real line

For a nontrivial measurable set on the real line, there are always exceptional points, where the lower and upper densities of the set are neither 0 nor 1. We quantify this statement, following work by V. Kolyada, and obtain the unexpected result that there is always a point where the upper and the lower densities are closer to 1/2 than to zero or one. The method of proof uses a discretized restatement of the problem, and a self-similar construction.

When Soap Bubbles Collide

Can you fill R^n with a froth of soap bubbles that meet at most n at a time? Not if they have bounded diameter, as follows from Lebesgue's Covering Theorem. We provide some related results and conjectures.

Lebesgue theorems in non-additive measure theory

In this paper, the sufficient and necessary conditions for different kinds of Lebesgue theorem in non-additive measure theory are presented, respectively. The equivalence among the Lebesgue theorem, the Monotone convergence theorems of fuzzy and of Choquet integral are shown. As direct results of the Lebesgue theorems, the necessary conditions for Egoroff's theorem are given.

Algebraic Formulation of Quantum Decoherence

An algebraic formalism for quantum decoherence in systems with continuous evolution spectrum is introduced. A certain subalgebra, dense in the characteristic algebra of the system, is defined in such a way that Riemann–Lebesgue theorem can be used to explain decoherence in a well defined final pointer basis.

A Short Proof of Lebesgue's Density Theorem

(2002). A Short Proof of Lebesgue's Density Theorem. The American Mathematical Monthly: Vol. 109, No. 2, pp. 194-196.

A Short Proof of Lebesgue's Density Theorem

A Short Proof of Lebesgue's Density Theorem

Order continuous of monotone set function and convergence of measurable functions sequence

In this paper, we introduce the concept of strongly order continuity of a set function, and use it to investigate the convergence of measurable functions sequence and of integral sequence on fuzzy measure space. Several equivalent conditions of strongly order continuity are presented. It is shown that a necessary and sufficient condition that Lebesgue's theorem remains valid for a monotone set function is that the set function possesses strongly order continuity.

On a functional equation arising from joint-receipt utility models

In this paper the functional equation¶¶\( F\Bigl(x + G(t(H(y - x))\Bigr) = S\Bigl(A(t)K(x),B(t)K(y)\Bigr), \qquad x,y \in I, x \le y, t \in [0,1] \),¶is solved under monotonicity and injectivity assumptions for the unkown functions involved. This equation turned out to be useful in the axiomatic treatment of various joint-receipt utility structures. After several reduction steps, the solution of the above equation is reduced to that of¶¶\( \Gamma^{-1}\bigl(\Gamma\bigl(\frac{H(u)}{H(u + v)}\bigr) \cdot \frac{H(x)}{H(x + u + v)}\bigr) = \frac{H(u)H(x + u + v)}{H(u + v)H(x + u)},\qquad x,u,v > 0,\,x + u + v\in I \).¶Assuming that H and \( \Gamma \) are strictly monotonic, it is shown that H is a log-concave function, and as a consequence, using Lebesgue's theorem, the everywhere-differentiability of \( \Gamma \) is deduced. Then, differentiating this equation, a functional differential equation for H is derived. The solutions of this new equation are then determined and the form of \( \Gamma \) is described. Thus, the general solution of the original equation is also obtained.

On the convergence of measurable set-valued function sequence on fuzzy measure space

In this paper we first discuss the measurable projection theorem on fuzzy measure space, and in this framework the characterization theorem with respect to measurability of a set-valued function is given. By means of the asymptotic structural characteristics of fuzzy measure, we discuss four forms of generalization for both Lebesgue's theorem, Riesz's theorem, and Egoroff's theorem, respectively. The relation between convergence of measurable set-valued function sequence and that of corresponding measurable real-valued function sequence are also discussed.

A Riemann Sum Upper Bound in the Riemann--Lebesgue Theorem

The Riemann--Lebesgue theorem is commonly proved in a few strokes using the theory of Lebesgue integration. Here, the upper bound $2\pi|c_k(f)|\le S_k(f)-s_k(f)$ for the Fourier coefficients ck is proved in terms of majoring and minoring Riemann sums Sk(f) and sf(k), respectively, for Riemann integrable functions f(x). This proof has been used in a course on methods of applied mathematics.

Lemme de Fatou pour l’integrale de Pettis

The purpose of this paper is to present Fatou type results for a sequence of Pettis integrable functions and multifunctions. We prove the non vacuity of the weak upper limit of a sequence of Pettis integrable functions taking their values in a locally convex space and we deduce a Fatou's lemma for a sequence of convex weak compact valued Pettis integrable multifunctions. We prove as well a Lebesgue theorem for a sequence of Pettis integrable multifunctions with values in the space of convex compact sets of a separable Banach space.

Convergence of sequence of measurable functions on fuzzy measure spaces

The purpose of this paper is to investigate the convergence of sequence of measurable functions on fuzzy measure spaces. Several classical results on the convergence of measurable functions, such as Egoroff's theorem, Lebesgue's theorem and Riesz's theorem, are extended to fuzzy measure spaces by using the pseudometric generating property and order-continuity of fuzzy measures.

Cyclic degree and cyclic coloring of 3-polytopes

A vertex coloring of a plane graph is called cyclic if the vertices in each face bounding cycle are colored differently. The main result is an improvement of the upper bound for the cyclic chromatic number of 3-polytopes due to Plummer and Toft, 1987 (J. Graph Theory 11 (1987) 505–517). The proof is based on a structural property of 3-polytopes, in a sense stronger than that implied by Lebesgue's theorem of 1940. Namely, precise upper bound is obtained for the minimum cyclic degree of 3-polytopes with the maximum face size at least 24. © 1996 John Wiley & Sons, Inc.

Uniform integrabblity and the lebesgue theorem on convergence in L 0-valued measures

We study integrals ∫fdμ of real functions over L 0-valued measures. We give a definition of convergence of real functions in quasimeasure and, as a special case, in L 0-measure. For these types of convergence, we establish conditions of convergence in probability for integrals over L 0-valued measures, which are analogous to the conditions of uniform integrability and to the Lebesgue theorem.

An Optimal Control Formulation of the Blaschke–Lebesgue Theorem

The Blaschke–Lebesgue theorem states that of all plane sets of given constant width the Reuleaux triangle has least area. The area to be minimized is a functional involving the support function and the radius of curvature of the set. The support function satisfies a second order ordinary differential equation where the radius of curvature is the control parameter. The radius of curvature of a plane set of constant width is non-negative and bounded above. Thus we can formulate and analyze the Blaschke–Lebesgue theorem as an optimal control problem.

Lattice-valued fuzzy measure and lattice-valued fuzzy integral

In this paper, (1) the concepts of lattice-valued fuzzy measure (with no valuation property) and lower (resp. upper) lattice-valued fuzzy integral are proposed, which give the unified discription to the fuzzy measures and fuzzy integrals studied by Delgado and Moral, Qiao, Ralescu, Adams, Sugeno, Wang, and Zhang; (2) some asymptotic structural characteristics of lattice-valued fuzzy measures are introduced, and some relations between them are given; (3) some concepts of convergences for lattice-valued functions are defined, and Riesz' theorem, Egoroff's theorems and Lebesgue's theorem for lattice-valued measurable function are proved; (4) the monotone increasing (resp. decreasing) convergence theorem and almost (resp. pseudo almost) everywhere convergence theorrem for lower (resp. upper) lattice-valued fuzzy integral are shown under some weak conditions.